Editing Dynamical system (section)

=== Geometrical definition ===
In the geometrical definition, a dynamical system is the tuple <math> \langle  \mathcal{T}, \mathcal{M}, f\rangle </math>. <math>\mathcal{T}</math> is the domain for time – there are many choices, usually the reals or the integers, possibly restricted to be non-negative. <math>\mathcal{M}</math> is a [[manifold]], i.e. locally a Banach space or Euclidean space, or in the discrete case a [[Graph (discrete mathematics)|graph]]. ''f'' is an evolution rule ''t''&nbsp;→&nbsp;''f''<sup>&nbsp;''t''</sup> (with <math>t\in\mathcal{T}</math>) such that ''f<sup>&nbsp;t</sup>'' is a [[diffeomorphism]] of the manifold to itself. So, f is a "smooth" mapping of the time-domain <math> \mathcal{T}</math> into the space of diffeomorphisms of the manifold to itself. In other terms, ''f''(''t'') is a diffeomorphism, for every time ''t'' in the domain <math> \mathcal{T}</math> .

==== Real dynamical system ====
A ''real dynamical system'', ''real-time dynamical system'', ''[[continuous time]] dynamical system'', or ''[[Flow (mathematics)|flow]]'' is a tuple (''T'', ''M'', Φ) with ''T'' an [[open interval]] in the [[real number]]s '''R''', ''M'' a [[manifold]] locally [[diffeomorphic]] to a [[Banach space]], and  Φ a [[continuous function]]. If Φ is [[continuously differentiable]] we say the system is a ''differentiable dynamical system''.  If the manifold ''M'' is locally diffeomorphic to '''R'''<sup>''n''</sup>, the dynamical system is ''finite-dimensional''; if not, the dynamical system is ''infinite-dimensional''. This does not assume a [[symplectic manifold|symplectic structure]]. When ''T'' is taken to be the reals, the dynamical system is called ''global'' or a ''[[Flow (mathematics)|flow]]''; and if ''T'' is restricted to the non-negative reals, then the dynamical system is a ''semi-flow''.

==== Discrete dynamical system ====
A ''discrete dynamical system'', ''[[discrete-time]] dynamical system'' is a tuple (''T'', ''M'', Φ), where ''M'' is a [[manifold]] locally diffeomorphic to a [[Banach space]], and  Φ is a function. When ''T'' is taken to be the integers, it is a ''cascade'' or a ''map''. If ''T'' is restricted to the non-negative integers we call the system a ''semi-cascade''.<ref>{{Cite book|title=Discrete Dynamical Systems|last=Galor|first=Oded|publisher=Springer|year=2010}}</ref>

==== Cellular automaton ====
A ''cellular automaton'' is a tuple (''T'', ''M'', Φ), with ''T'' a [[lattice (group)|lattice]] such as the [[integer]]s or a higher-dimensional [[integer lattice|integer grid]], ''M'' is a set of functions from an integer lattice (again, with one or more dimensions) to a finite set, and Φ a (locally defined) evolution function.  As such [[cellular automata]] are dynamical systems. The lattice in ''M'' represents the "space" lattice, while the one in ''T'' represents the "time" lattice.

==== Multidimensional generalization ====
Dynamical systems are usually defined over a single independent variable, thought of as time. A more general class of systems are defined over multiple independent variables and are therefore called [[multidimensional systems]]. Such systems are useful for modeling, for example, [[image processing]].

==== Compactification of a dynamical system ====
Given a global dynamical system ('''R''', ''X'', Φ) on a [[locally compact]] and [[Hausdorff space|Hausdorff]] [[topological space]] ''X'', it is often useful to study the continuous extension Φ* of Φ to the [[one-point compactification]] ''X*'' of ''X''. Although we lose the differential structure of the original system we can now use compactness arguments to analyze the new system ('''R''', ''X*'', Φ*).

In compact dynamical systems the [[limit set]] of any orbit is [[non-empty]], [[compact space|compact]] and [[simply connected]].